Keten and Buehler Reply
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Keten, Sinan , and Markus J. Buehler. “Keten and Buehler
Reply:.” Physical Review Letters 102.12 (2009): 129802. © 2009
The American Physical Society.
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http://dx.doi.org/10.1103/PhysRevLett.102.129802
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American Physical Society
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Final published version
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http://hdl.handle.net/1721.1/51334
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Detailed Terms
PRL 102, 129802 (2009)
PHYSICAL REVIEW LETTERS
Keten and Buehler Reply: We recently proposed a fracture mechanics model to predict an upper limit for the
rupture strength of a polypeptide chain stabilized by
H bonds at vanishing pulling rates [1]. In the preceding
Comment [2], Makarov raises the concern that in the shear
loading scenario discussed in [1], the force prediction
should actually be infinite. Here, we emphasize that our
original derivation is correct and point out the differences
between our approach and that presented in [2]. According
to Eq. (1) in [2], the free energy of the system can be
written as Aðx; Þ ¼ AWLC ðx=Þ Fðx Þ s ðL Þ. This requires that the attached polypeptide chain is
fully extended, which differs from our assumptions [1].
Our assumption is that the bonded chain is fully relaxed, in
which case in the force term in Eq. (1) in [2] disappears,
leading to Aðx; Þ ¼ AWLC ðx=Þ Fx s ðL Þ.
This assumption is evident both from the fact that free
energies are consistently calculated
with respect to the
R
fully relaxed state [AWLC ¼ 0 FWLC ðÞd] and from
the expression for the work done by the external force. In
[1], the variable dx corresponds to the displacement of the
chain end, and the integration is performed with respect to
a point that remains stationary. While our assumption may
be considered limiting, it is not methodologically incorrect
as argued in [2]. On the contrary, our original model [1]
represents a minimal theory that has the least number of
parameters and yields the simplest analytical formulation
possible.
The underlying assumptions between Makarov’s approach and our derivation represent two extreme cases.
In general, the attached segment of the chain would have
a finite average end-to-end distance given as sL, where 0 <
s 1 and L is the contour length of the attached segment
(see inset, Fig. 1). Then Eq. (1) in [2] becomes Aðx; Þ ¼
AWLC ðx=Þ Fðx sÞ Rs ðL Þ þ ðL ÞAFIX ðsÞ
where the last term AFIX ¼ s0 FWLC ðÞd has to be introduced since free energy integrals are taken with respect
to the zero-stretch state, whereas the liberated segment
already has a prestretch (noting that we treat entropic
elasticity and H bond terms separately). If this consideration were included in Makarov’s expression, it would yield
a rupture strength value of zero as the WLC model diverges
at full extension. Quantifying the strength with assumptions on initial extension state of the bonded chain is thus
clearly not straightforward.
Proceeding as we did previously [1] but with the prestretch accounted for, we substitute ¼ x= and solve
for the critical that satisfies AWLC ðcr Þ þ Fðs cr Þ þ
s AFIX ðsÞ ¼ 0,
FWLC ðcr Þ ¼ Fcr .
For
Eb ¼
4 kcal=mol, p ¼ 0:4 nm, variations of s in the range 0 s 0:6 yield results within the error range of results
shown in Fig. 4 in [1]. The case s ! 1 (specific to
beta-sheet systems) would involve the possibility of
stick-slip motion as a key failure mechanism (H bonds
0031-9007=09=102(12)=129802(1)
week ending
27 MARCH 2009
FIG. 1 (color online). Strength prediction for varying values of
s, including an extrapolation to s ¼ 1 (double stranded slip
case). The linear extrapolation is done for 0 s 0:8 (values
admissible to the WLC model). The value for s ! 1 approaches
the prediction by our original model [1], illustrating the robustness of the main contribution of our Letter with respect to
different physical boundary conditions. The inset shows the
double strand shear condition and parameter s, defined as the
ratio of end-to-end length to contour length for the attached
segment.
reform after sliding). We evaluate the critical condition
as 2AWLC ðcr Þ þ Fðs 2cr Þ þ s 2AFIX ðsÞ ¼ 0 and
extrapolate results to the case s ! 1 (Fig. 1). These results
are in excellent agreement with our original prediction,
illustrating the robustness of the results reported in [1].
Specifically, our results shown in Fig. 1 here and Fig. 4 in
[1] (both lead to 127 pN) agree well with experiments
(Fig. 1 in [1]). Typical rupture forces (the case considered
here, distinct from refolding) in force-quenching experiments [3] agree with our predictions. The number of
H bonds present and the loading geometry will affect the
strength of the assembly as discussed in [1]. The predictions reported in [1] correspond to an upper strength limit
of a large uniformly loaded cluster of H bonds.
Sinan Keten and Markus J. Buehler
Laboratory for Atomistic and Molecular Mechanics
Department of Civil and Environmental Engineering
Massachusetts Institute of Technology
77 Massachusetts Avenue, Room 1-235A&B
Cambridge, Massachusetts, USA
Received 13 October 2008; published 26 March 2009
DOI: 10.1103/PhysRevLett.102.129802
PACS numbers: 82.37.Rs, 62.20.M, 81.40.Lm, 87.15.La
[1] S. Keten and M. J. Buehler, Phys. Rev. Lett. 100, 198301
(2008).
[2] D. E. Makarov, preceding Comment, Phys. Rev. Lett. 102,
129801 (2009).
[3] S. Garcia-Manyes et al., Biophys. J. 93, 2436 (2007).
129802-1
Ó 2009 The American Physical Society